In this section, we give a didactic treatment of our approach to motivate the notion of total variation stability.
Consider neighbouring datasets \(S\)  and \(S^{\prime }\)  and let \(P = (S)\)  and \(Q=(S^{\prime })\)  for some randomized algorithm \(\) . The algorithm first computes on \(S\) , and then observes edit requests which generate \(S^{\prime }\)  as the current dataset. To satisfy exact unlearning , we need a procedure which moves \(P\)  to \(Q\) .
This is akin to the well-studied optimal transport problem [1]}, which briefly explain below.
Given probability distributions \(P\)  and \(Q\)  over measurable space \(\) , and a cost function \(c: \times \rightarrow \) , the goal is to transport from \(P\)  to \(Q\)  using the minimum cost.
Formally, let \(\Pi (P, Q)\)  denote the set of couplings (or transport plans) of \(P\)  and \(Q\) ; the modern (Kantorovich's) formulation asks for a transport plan \(\pi \)  which minimizes the expected cost: \( \min _{\pi \in \Pi (P,Q)} \mathbb {E}_{(x,y)\sim \pi } c(x,y)\) .
